Persistent obstruction theory for a model category of measures with applications to data merging

Collections of measures on compact metric spaces form a model category (“data complexes”), whose morphisms are marginalization integrals. The fibrant objects in this represent collections which there is measure product space that marginalizes to any pairs its factors. homotopy and homology for allow measurement obstructions finding larger spaces. obstruction theory compatible with filtration built from the Wasserstein distance measures. Despite abstract tools, motivated by widespread problem data science. Data complexes provide mathematical foundation semi-automated data-alignment tools common commercial database software. Practically speaking, shows JOIN operations subject genuine topological obstructions. Those can be detected an cocycle resolved moving through filtration. Thus, collection databases has persistence level, difficulty JOINing those databases. Because general formulation, persistent also encompasses multi-modal fusion problems, some forms Bayesian inference, probability couplings.